For a sufficiently nice topological space, the fundamental group at a point can be reconstructed as a group of deck transformations of the universal covering space, which is the same as the automorphisms of the fiber over that point of the projection map. The deck transformations are monodromies induced by loops at the base point. The functor which assigns to a point the fiber functor over it, generalizes to fiber functors in the Tannakian formalism of Grothendieck which defines in more general setups the fundamental groupoid as the group of automorphisms of the appropriate fiber functor. See also fundamental group of a topos.
covering spaces of $X$ : $\pi_1(X)$-sets
for a locally path connected, semilocally simply connected? topological space $X$.
The objects on the left are not difficult to define for schemes (at least naively – one really needs trivialisations over étale covers), but it may not be entirely immediate what the fundamental group defined in terms of loops should be.
The reason Galois’s name is attached to this theory is that in the case of the scheme $Spec(k)$, the objects corresponding to the covering spaces are simply field extensions $Spec(k')$. The fundamental group of schemes defined in this way is the algebraic fundamental group, and is a profinite group.
The basic idea of Grothendieck’s Galois theory may be extended to objects in an topos – leading to a notion of fundamental group of a topos – and then further to objects in any (∞,1)-topos. For more on this see homotopy group of an ∞-stack.
Given an arrow $f:x \to y$ in a category $C$ the category of arrows compatible with $f$, denoted $Comp(f)$ is the full subcategory of the undercategory $x \downarrow C$ on the arrows that coequalize the same pairs $g,h:w\rightrightarrows x$ that $f$ does.
An arrow $f:x\to y$ in a category $C$ is a strict epimorphism if it is initial in $Comp(f)$.
It is not obvious, but a strict epimorphism is an epimorphism.
In what follows, Let $C$ be a category and $F:C \to Set$ a functor. The axioms presented here are as in
J. P. Murre, Lectures on an introduction to Grothendieck’s theory of the fundamental group, Tata Inst. of Fund. Res. Lectures on Mathematics 40, Bombay, 1967. iv+176+iv pp.
and copied also in
Some terminology: $X\in C$ is called finite if $F(X)$ is a finite set. Let $\int_F C$ denote the category of elements of $F$, in which an object $(X,a)$ is called finite if $X$ is finite.
G0) The full subcategory of $\int_F C$ on the finite objects is cofinal.
G1) $C$ has all finite limits
G2) $C$ has an initial object, finite coproducts and quotients by finite groups
G3) Given $f:x\to z$ in $C$ there is a factorisation $x \stackrel{e}{\to} y \stackrel{i}{\to} z$ where $e$ is a strict epimorphism and $i$ is a mono. Also, $y$ is assumed to be a direct summand of $z$.
G4) $F$ preserves finite limits
G5) $F$ preserves initial object, finite sums, quotients by finite group actions and sends strict epimorphisms to surjections
G6) $F$ reflects isomorphisms
The functor $F$ is called the fibre functor, and the pair $(C,F)$ is sometimes called a Galois category.
It follows from the axioms that $F$ is a pro-representable functor. The automorphism group of the pro-object $P$ representing $F$ is (should be. I’m not familiar enough with pro-objects) a profinite group $\pi$. This acts on $F(X) = [P,-]$ by precomposition (talking out of my depth here – it’s getting a bit vague) and so $F$ lifts to a functor to $\pi-Set$, and Grothendieck’s result is that this functor is an equivalence of categories.
There are several modifications one can make the above. In the case that $C$ is the category of covering spaces of a nice enough space, the functor $F$ is representable by the universal covering space, and so there is a ‘representable’ version of the above, not needing to utilise profinite groups. One can also consider just the connected-objects version, and end up with an equivalence to the category of transitive $\pi$-sets.
Even for the classical case of the inclusion of fields, Grothendieck’s Galois theorem gives more general statement than the previously known. This is the Grothendieck’s version of the Galois correspondence theorem for fields:
Let $K \subset L$ be a finite dimensional Galois extension of fields. Let $Gal[L : K]$ denote the group of $K$-automorphisms of $L$ and $Gal [L : K]-finSet$ the category of finite $Gal[L : K]$-sets. By $SplitfinAlg_K(L)$ denote the finite dimensional $K$-algebras which are split over $L$; here $L$ itself is an object. Consider the representable functor $h_L = Hom_K(-,L):SplitfinAlg_K(L)\to Set$. It takes values in the subcategory of finite sets and it comes with a canonical $Gal[L : K]$-action. In other words, this functor factors through $Gal [L : K]-finSet$. Moreover, the corresponding functor
is an equivalence of categories.
There is an infinitary version as well, generalizing the classical Galois theorem on infinitary Galois extensions.
Thus let $K\subset L$ be an arbitrary Galois extension. Now $Gal[L:K]$ denotes the profinite Galois group and $Gal[L:K]-profinSpace$ the category or profinite $Gal[L:K]$-spaces. $SplitAlg_K(L)$ denotes the category of $K$-algebras split over $L$ (possible infinite-dimensional). Then there is a canonical anti-equivalence of categories
(factorizing a profinite-space version of the representable functor $Hom_K(-,L)$).
A special case of this is the following: the category of étale k-schemes reps. étale group schemes? for a field $k$ is equivalent to the category of sets equipped with an action of the absolute Galois group reps. to the category of Galois modules of the absolute Galois group.
Let $E$ be a Grothendieck topos. Then there exist an open localic groupoid $G$ such that $E$ is equivalent to the category of étale presheaves over $G$. One of the classical references is
This is a variant of the theorem in the setting of locales from
The original development of the theory by Grothendieck is in SGA1.
A more recent treatment can be found in
and more related categorical and topos theoretic aspects in
A very approachable account is given in
(This has the advantage of looking towards Grothendieck’s dessins d’enfants.)
Basic intuition is explained in
The construction for general toposes is described in section 8.4 of
and, a current state of the art description is in
A modern approach from classical via Grothendieck up to categorical Galois theory based on precategories and adjunctions is in
The application of a general Tannakian theorem of Saavaedra Rivano, as corrected by Deligne, to the “differential Galois theory” for differential instead of algebraic equations is in the last chapter of Deligne’s Catégories Tannakiennes.